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Magnetically Induced Rotating Rayleigh-Taylor Instability
06:42

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Published on: March 3, 2017

Eckhaus instability and homoclinic snaking.

A Bergeon1, J Burke, E Knobloch

  • 1IMFT UMR CNRS 5502-UPS UFR MIG, 31062 Toulouse Cedex, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 13, 2008
PubMed
Summary
This summary is machine-generated.

Homoclinic snaking, a phenomenon in bistable systems, describes oscillating localized states. This study examines how snaking terminates in finite domains, relating it to Eckhaus instability.

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Area of Science:

  • Nonlinear dynamics
  • Pattern formation
  • Fluid dynamics

Background:

  • Homoclinic snaking involves oscillations of spatially localized states in bistable systems.
  • This behavior is observed in the subcritical Swift-Hohenberg equation and doubly diffusive convection.
  • In unbounded systems, snaking can continue indefinitely, resembling periodic states.

Purpose of the Study:

  • To investigate the termination of homoclinic snaking in finite and periodic domains.
  • To identify factors influencing termination points of snaking branches.
  • To explore the relationship between snaking termination and Eckhaus instability.

Main Methods:

  • Analysis of homoclinic snaking in spatially extended systems.
  • Examination of the subcritical Swift-Hohenberg equation and doubly diffusive convection.
  • Investigating the transition to spatially periodic states and their instabilities.

Main Results:

  • Snaking branches terminate when the localized state length approaches the domain size.
  • Termination points are influenced by domain size and system parameters.
  • The termination is linked to the Eckhaus instability of the spatially periodic state.

Conclusions:

  • Homoclinic snaking in finite domains exhibits turnover behavior.
  • Understanding termination mechanisms is crucial for predicting pattern formation in confined systems.
  • The study provides insights into the interplay between localized structures and global instabilities.